Papers

We investigate a doubly degenerate nutrient-taxis system of the form \begin{equation*} \begin{cases} u_t = \nabla \cdot (u v \nabla u) - \chi \nabla \cdot (u^\alpha v \nabla v) + \ell u v, \qquad &x \in \Omega, \ t > 0, v_t = \Delta v - u v, \qquad &x \in \Omega, \ t > 0, \end{cases} \end{equation*} subject to the homogeneous Neumann boundary conditions in a smoothly bounded convex domain $\Omega \subset \mathbb{R}^n$ with $n\in \left \{ 3,4,5 \right \}$, where $\alpha \geq 1$, $\chi>0$ and $\ell \geq 0$. For any suitably regular initial data, we establish the global existence of a weak solution that remains uniformly bounded in time, provided that $\alpha$ lies in the range $\left[1, \frac{5}{2} - \frac{n}{4}\right)$, and we also determine the large-time behavior of these solutions. Our proof relies on several novel functional inequalities, a bootstrap argument, and a Moser iteration method.

The present work investigates the asymptotic behaviours at the zero-noise limit of the first near collision-time and first near collision-location between a pair of independent $d$-dimensional Brownian-driven self-stabilizing (McKean-Vlasov type) diffusions. These asymptotic are considered in a peculiar setting where the systems evolve in a bi-stable landscape and collisions are only triggered by the combined action of the Brownian noises. As the Brownian perturbations fade away, we show that the near collision-time increases at an explicit exponential rate and that related collision-locations persist in specific regions of the space. These results are mainly derived by tailoring classical Freidlin-Wentzell's exit-time (and exit-location) estimates into collision estimates. Similar asymptotic are established for related mean-field interacting particle system approximation, and for the one-dimensional case (where true collisions can be examined

The compressible Euler–Riesz equations arise in the modelling of a wide range of physical phenomena, including stellar dynamics, plasma physics, and mathematical biology. We study rotating steady states of the attractive compressible Euler–Riesz equations, which we call rotating Riesz stars, and establish existence and nonlinear stability or instability results in both the mass-subcritical and mass-supercritical regimes. In the mass-subcritical regime, we prove the existence of rotating Riesz stars under suitable subhomogeneity assumptions on the angular momentum profile. We then establish their nonlinear stability through a concentration compactness argument adapted to the axisymmetric setting. Rotation creates new compactness difficulties, most notably the possibility that minimising sequences are tight along rings whose radii diverge to infinity. In the mass-supercritical regime, we prove existence in the polytropic setting under suitable superhomogeneity assumptions on the angular momentum profile, thereby extending the theory beyond the small angular velocity regime. The proof requires a careful analysis of mass-preserving scalings, which are more delicate than in the non-rotating case. Finally, by analysing the concavity of the free-energy along these scalings, we establish the instability of the resulting mass-supercritical rotating Riesz stars. Our results show that rotation can have either a stabilising or a destabilising effect, depending on the singularity of the Riesz interaction.

We investigate the stationary Dirac equation associated with a generalized Kronig-Penney model in (N+1)-dimensional Clifford analysis. Away from the interaction sites, the free Dirac system is reformulated by introducing suitable generalized Cauchy-Riemann operators, leading to a coupled first-order system for two Clifford-valued fields. By means of characteristic hypercomplex variables, this system is reduced to a pair of decoupled generalized Helmholtz equations. To construct explicit solutions, we introduce a bivariate Clifford-Appell polynomial basis adapted to the generalized Cauchy-Riemann operators. The Appell expansions transform the differential problem into an algebraic recurrence for the expansion coefficients, yielding explicit series representations of the solutions. The proposed framework provides a unified Clifford-analytic formulation of the free propagation problem and establishes an explicit connection between generalized Helmholtz equations, and Appell polynomial systems in hypercomplex function theory.

Large language models (LLMs) have emerged as a powerful tool for automated evolutionary optimization, but existing methods remain limited in pattern reuse, error-aware refinement, and retrieval robustness across diverse tasks. To address these limitations, we propose OptGraph, the first optimization agentic workflow that introduces graph retrieval-augmented generation (GraphRAG). Specifically, OptGraph first constructs reusable experience as a typed graph, capturing the relationships among modeling patterns, problem formalization, implementation details, and error corrections. In the inference stage, OptGraph leverages graph neighborhood information to enrich retrieved knowledge, providing structured context to improve modeling, verification, and iterative refinement. Moreover, OptGraph supports adaptive knowledge updates, enabling the distillation of execution traces and verification feedback into reusable graph knowledge without ndertaking LLM parameter tuning. Extensive experiments on benchmark datasets show that our proposed OptGraph achieves an average exact accuracy 8.9% higher than the state-of-the-art prompt-based automated optimization frameworks. Our code has been made available at https://github.com/xianchaoxiu/OptGraph.

For an $n$-vertex graph $G$, let $N(H_3,G)$ be the number of injective labeled copies of the red-blue path $H_3$ for which the two blue pairs are mapped to non-edges of $G$ and the red pair is mapped to an edge of $G$. We determine the maximum limiting value of $N(H_3,G)/n^4$ and give an extremal construction, which is the disjoint union of a clique and an asymptotically regular graph. The proof uses weighted vertex quotients and degree-square tie-breaking. We thereby resolve the exceptional four-vertex case left open in the recent classification of non-complete red-blue graphs.

This paper studies the generalized multi-source Weber problem with set-valued targets in the framework of minimal time functions. We first establish the existence of global and local optimal solutions and investigate several qualitative properties of the corresponding solution sets, including closedness, compactness, and conditions ensuring boundedness or unboundedness. Next, we derive explicit Lipschitz continuity properties for the objective function and the associated optimal value function with respect to perturbations of the target sets. We then introduce the global and local solution mappings and study their stability properties from the viewpoint of set-valued analysis. These results provide a quantitative and qualitative sensitivity analysis for the generalized multi-source Weber problem in the setting of minimal time functions.

We propose a hybrid planning method for deriving prognostics-based predictive maintenance policies. The method accounts for the available decision options, the information on the future state of the system provided by a prognostic model, and the costs of the underlying renewal-reward process. It results in policies defined by only a few parameters, which can be determined based on theoretical considerations or by optimization from run-to-failure data. We demonstrate the potential of the method in two separate predictive maintenance decision settings: preventive replacement and preventive ordering. Numerical investigations show that the derived policies rival the performance of optimized benchmark policies, while being significantly more efficient and robust against overfitting.

We study the aperiodic inverse-limit system constructed in our earlier work as an example of a finite-mean-dimensional dynamical system without the marker property. We prove that its mean dimension and Meyerovitch's dynamical dimension are both equal to $N$. Despite the absence of the marker property, the system admits an equivariant topological embedding into the cubical shift with alphabet dimension $3N+2$. As an auxiliary result, we prove that the dynamical dimension of the full shift over any compact metrizable alphabet is exactly the covering dimension of the alphabet, including when this dimension is infinite.

We study quantum analogues of Bayesian networks on a directed acyclic graph (DAG), distinguishing two constructions for positive definite density operators on finite-dimensional tensor-product Hilbert spaces. The intrinsic construction starts from a joint state and its conditional-independence properties. The extrinsic construction assembles a state sequentially from prescribed local quantum kernels, following an ordering compatible with the arrows of the DAG. For the intrinsic construction, we prove the equivalence of the ordered, local, and global directed Markov properties, together with entropy, recursive-factorization, and logarithmic characterizations. The extrinsic construction always gives a normalized state and recovers each kernel as a conditional on all preceding systems. The same kernel, however, need not be recovered from the marginal on the vertex and its parents; a three-qubit example exhibits this obstruction. We prove that independence of the chosen topological ordering is sufficient exactly for transitive DAGs: every order-invariant kernel family then yields an intrinsically directed Markov state. Finally, we associate a logarithmic candidate with every positive definite state and DAG, prove that it is subnormalized, and show that the trace-one candidate is a directed Markov state. Both the candidate and the excess global information are invariant under DAG Markov equivalence.

A binary operation $\cdot$ which satisfies the identity $(x \cdot y) \cdot x = y$ is called a semisymmetric quasigroup. We show that the nucleus of a semisymmetric quasigroup is either empty or an elementary abelian 2-group coinciding with the centre, and that a semisymmetric quasigroup with a non-empty nucleus is necessarily a Mendelsohn loop, i.e. the loop associated with a Mendelsohn triple system. We derive necessary and sufficient conditions for the existence of a semisymmetric quasigroup of order $n$ with nucleus of order $m$. Furthermore, we characterize the nuclear elements of a Mendelsohn loop in terms of a particular orientation of the Pasch configuration in the associated triple system.

Let $F$ be an $r$-vertex graph. In this paper, we study the $F$-factor problem in random induced subgraphs of dense graphs. We show that for any $r$-vertex graph $F$ and $\gamma>0$, if $H$ is an $n$-vertex graph with minimum degree at least $(1-1/\chi_{cr}(F)+\gamma)n$, then for every fixed $p \in (0,1)$, the random induced subgraph $H[p]$ contains an $F$-factor with probability at least $1/(rq)-o_n(1)$, where $q\in \mathbb{N}$ is the order of certain coset group defined from $H$. The probability is asymptotically best possible for infinitely many $F$ and $H$ and yields that a $1/(rq)-o_n(1)$ proportion of the subsets of $H$ induce $F$-factors, interestingly, regardless of whether $H$ itself admits an $F$-factor. Similar results are obtained for perfect matchings in hypergraphs under minimum degree conditions. Our proof combines concentration inequalities, lattice point counting in $\mathbb{Z}^d$ and structural theorems for $F$-factors in dense (hyper)graphs.

We give a recursive reformulation and extension of the independent set of size $134753$ in $C_7^{10}$ constructed by N. Itty, C. D. Rosin, C. Carstensen, and D. Reichman (arXiv:2607.21517v1). We prove a product lemma that combines gadgets of different dimensions while preserving the required independence conditions. Starting from the size-$367$ independent set in $C_7^5$ of S. C. Polak and A. Schrijver (Information Processing Letters 143 (2019), 37-40), the construction gives an explicitly specified independent set in $C_7^{200}$. Consequently, \[ \Theta(C_7)\geq 3.2587891539086910161967650155\ldots . \] An accompanying program verifies the finite assertions about the five-dimensional base gadget and performs the exact integer computations used in the recursion.

This paper investigates the strong structural controllability of multi-agent networks. Based on the definition of equitable partitions, an upper bound for the strong structural controllable subspace (SSCS) is established. To reflect the physical significance of matrix weights where the state dimension is greater than one, the multi-agent system is modeled using higher-order dynamics. Furthermore, to address matrix singularity and asymmetric couplings, a matrix space basis decomposition method is proposed to transform the matrix-weighted network into layered scalar networks. Additionally, by extending this basis decomposition to the lower bound estimation, a layer-specific distance partition (LDP) is introduced. This formulation establishes a tighter Squeeze Theorem, narrowing the mathematical boundaries for the controllable subspace by capturing layer-specific structural delays. To systematically identify the optimal basis that minimizes the bounds gap, an algebraic algorithm based on null-space projection is formulated. Furthermore, by introducing pattern matrices and generic rank, the almost-everywhere existence of this optimal basis in the parameter space is rigorously proved, perfectly aligning with the definition of strong structural controllability. To break the NP-hard combinatorial bottleneck of manually pre-defining the targets, a polynomial-time automated discovery algorithm based on the multi-layer Weisfeiler-Lehman (WL) color refinement is proposed. Finally, the strong structural observability and invariant attributes of the network are evaluated. Numerical examples with asymmetric matrix weights and directed multi-layer topologies are provided to verify the derived theorems.

A hole in a graph is an induced cycle of length at least $4$. A cap is a hole together with a vertex adjacent to exactly two consecutive vertices of it. Chen, Xu and Xu conjectured that if $q\ge2$ and $G$ is a $(\mathrm{cap},\mathrm{even\ hole})$-free graph with no odd hole of length at most $2q-1$, then $\chi(G)\le \left\lceil \frac{2q+1}{2q}\omega(G)\right\rceil.$ They confirmed the conjecture for $q \le 3$. In this paper, we prove the conjecture for all $q \ge 3$. As a corollary, we prove that for such a graph $G$, $\chi_f(G)\le \frac{2q+1}{2q}\omega(G).$

We study gabi-monads on skew-closed categories, extending the gabi-algebras of Berger, the second author, and Vercruysse beyond the linear case. Our main reconstruction theorem identifies gabi-monad structures on a monad with skew-closed structures on its Eilenberg–Moore category for which the canonical forgetful functor is strict closed. We compare this notion with closed monads in the sense of Kock, showing that in representation-theoretic cases these notions are quite different. On closed monoidal categories, every left Hopf monad is a normal gabi-monad, but the converse fails in general. We characterise when a gabi-monad is Hopf by the invertibility of the corresponding parametric mates, which recovers the ring-theoretic result that normal gabi-algebras over a commutative base ring are Hopf algebras. The theory of gabi-monads admits several natural examples, such as torsion-free modules, reflexive digraphs, and simplicial complexes, that we will explore in detail; we also study pointed sets as a quasi-example.

Let $g(n,k)$ be the maximum number of distinct sizes of maximal cliques in an $n$-vertex $k$-uniform hypergraph, and let $f(n,k)=n-g(n,k)$. We determine the asymptotic order of $f(n,k)$ for every fixed integer $k\ge 3$. Define $L_2(x)=\max\{2,\log_2(\max\{1,x\})\}$, and, for $j\ge 3$, let $L_j(x)$ be the least number of iterations of $L_{j-1}$ needed to reach a value at most $16$. We prove that $$ f(n,k)=\Theta_k(L_k(n)).$$ In particular, $f(n,3)=\Theta(\log^{*}n)$, where $\log^{*}n$ denotes the iterated logarithm. We also determine the asymptotic behaviour of the layered-tree threshold $c(n,k)$ arising from Gao's insertion-tree method: $$ c(n,k)=\log_2 L_k(n)+O_k(1). $$ Consequently, $f(n,k)=\Theta_k\!\left(2^{c(n,k)}\right)$. Our result gives a negative answer to Gao's question in the case $k=3$.

We develop a selfnormalized approach to inference for relevant changes in functional time series measured by the supremum norm. The main difficulty is that the supremum norm is not Hadamard differentiable, so standard projection-based selfnormalization does not apply and the limiting distribution may depend on the geometry of the extremal set and the long-run covariance structure. We address this problem by replacing the supremum norm with a smooth log-sum-exp approximation and constructing a projected selfnormalizer from its derivative. The resulting statistic has an asymptotically pivotal distribution that is free of long-run covariance nuisance parameters and depends only on the break location. We derive explicit smoothing-bias expansions for both isolated nondegenerate extrema and extremal sets of positive measure. To avoid direct estimation of geometric quantities such as the number, curvature, or measure of the extrema, we combine several smoothing levels to cancel the leading bias terms. This yields an asymptotically exact test for relevant changes under mild regularity conditions. More generally, the proposed smoothing and bias-correction principles provide a framework for combining selfnormalization with supremum-type statistics in problems involving relevant hypotheses.

We introduce and study the perturbed $\beta$-corners process, a deformation of the classical $\beta$-corners process. The latter is a probability measure on interlacing arrays of real numbers that, for the classical values $\beta=1,2,4$, describes the joint distribution of eigenvalues of principal submatrices of a Gaussian random matrix with the corresponding GOE/GUE/GSE symmetry. For classical $\beta$, the perturbed process arises from adding a deterministic diagonal matrix $A=diag(a_1,...,a_N)$ to a GOE/GUE/GSE matrix. The eigenvalues of the resulting random matrix depend symmetrically on $a_1,...,a_N$, but this symmetry does not extend to the whole corners process. We extend the construction to all $\beta>0$ via multivariate Bessel functions, and analyze the crystallization ($\beta\to\infty$) of the resulting interlacing array, proving a law of large numbers and a central limit theorem in two regimes. For fixed perturbation, the eigenvalue repulsion dominates, and the array crystallizes on the same lattice of polynomial-derivative roots, with the same discrete Gaussian free field fluctuations, as in the unperturbed case treated by Gorin and Marcus (arXiv:1706.07393). New phenomena appear in the second regime, where the $a_i$'s grow linearly in $\beta$. Then the external source competes with the repulsion at leading order. Here the array freezes on a deformed lattice characterized by a coupled system of optimality equations. The fluctuations are governed by the same discrete Gaussian free field, now attached to the deformed lattice. The deformed lattice equations decouple in the special case of a single spike in the last coordinate. Then the deformed lattice is obtained explicitly by applying one shifted derivative $D_c f = f' + cf$ followed by iterated ordinary derivatives.

The \textit{$\alpha\beta$-Tessarine Toolbox} is introduced as a high-performance MATLAB framework for generalized hypercomplex tensor algebra. By implementing the $\alpha\beta$-tessarine mathematical structure, a strictly more general environment than existing quaternion or limited tessarine packages is provided. This open-source software bridges abstract theoretical models and experimental applications, allowing researchers to efficiently solve multidimensional problems. Built-in capabilities include exact hypercomplex matrix factorizations, such as Singular Value Decomposition (SVD) and QR, optimized for large-scale image in-painting, denoising, and digital watermarking. The scalable architecture facilitates new computational research in machine learning and multidimensional signal processing.

In computational anatomy, analyzing morphological variability across shape populations often requires multi-component deformation models that combine structured motions and unconstrained diffeomorphisms. However, a major challenge arises during the registration process, as high-dimensional deformations tend to absorb lower-dimensional components, altering the true geometric variability and preventing accurate statistical analysis. To address this issue, we introduce a novel coupling score designed to decouple distinct deformation modes during registration. This score is defined using first variation of varifolds which is a varifold representation of the infinitesimal action of vector fields on shapes. The proposed score quantifies the extent to which the action of a given vector field on a shape can be replicated by another subspace of vector fields. We provide a theoretical analysis of this coupling score, illustrating its behavior on finite-dimensional spaces. Finally, we integrate this score as a penalization term in registration problems. Numerical experiments illustrate its efficiency in various use cases such as enforcing or preventing specific motions, and iteratively correcting complex matching scenarios by enforcing structured directional priors.

A connected graph G is said to be a bisplit graph if the vertex set of G can be partitioned into a stable set and a complete bipartite graph. We establish the following dichotomy with chordality being the parameter; for chordal bisplit graphs, Hamiltonian cycle (HCYCLE) and Hamiltonian path (HPATH) problems are polynomial-time solvable, and for chordal bipartite bisplit graphs, HCYCLE (HPATH) is NP-complete. We further strengthen the result of [1] and show that HCYCLE (HPATH) is polynomial-time solvable on P5-free chordal bipartite graphs (bipartite chain graphs) and NP-complete on P10-free chordal bipartite graphs. By using our polynomial results on HCYCLE (HPATH) as a framework, we solve many variants and generalizations of HCYCLE (HPATH), which are also reported in this paper.

A general framework for deriving characteristic functions for reproducing kernels that do not necessarily possess the complete Pick property was recently established by Bhattacharyya and Jindal. We show that, in this setting, the existence of a characteristic function is equivalent to a Beurling-type invariant subspace condition. Combined with recent results characterizing kernels satisfying this condition, our theorem implies that the existence of a characteristic function is equivalent to a concrete Agler-type decomposition of the underlying kernels.

In 1998, Jorgensen and Pedersen constructed the first example of a singular continuous spectral measure. Precisely, they proved that the self-similar measure generated by the iterated function system $\{\frac{x-1}{4},\frac{x+1}{4}\}$ with equal weights, denoted by $\mu_{1/4}$, is a spectral measure with a spectrum $\Lambda_4$, called the canonical spectrum,\[ \Lambda_4 := \set{ \sum_{j=0}^{m-1}\varepsilon_j4^j: m\ge 1,\ \varepsilon_j\in\{0,1\} }. \] For $f\in L^1(\mu_{1/4})$, let $S_n f$ be the $n$-th partial sum of its Mock Fourier series with respect to $\Lambda_4$. We prove that the associated maximal operator $S^{\ast}f=\sup_n|S_n f|$ is of weak type $(1,1)$. Consequently, $S_n f\to f$ $\mu_{1/4}$-almost everywhere on $\supp(\mu_{1/4})$. This solves a long-standing open problem of Strichartz \cite[p.~341]{Str06}.

We present a framework combining discrete gradient (DG) methods with the Scalar Auxiliary Variable (SAV) approach to construct structure-preserving integrators for dissipative and conservative systems. The key observation is that SAV quadratization lifts the dynamics to an extended state space on which the modified energy has an exact discrete-gradient identity. This viewpoint yields three integrators with different accuracy and cost profiles: a first-order semi-implicit Forward Euler scheme, a second-order self-adjoint Midpoint scheme, and a second-order Predictive scheme with reduced implicitness. The construction extends to almost-Poisson systems and preserves selected Casimir invariants under an enforceable discrete condition. Numerical experiments cover the Allen–Cahn equation, an Ohta–Kawasaki-type nonlocal gradient flow, a double-well Hamiltonian oscillator, and a Poisson system with a nonlinear cubic Casimir.

We consider level lines of Gaussian free field (GFF) in annulus with alternating boundary conditions. We calculate the probability that all level lines cross the annulus. Such probability is given by the ratio between two partition functions. These two partition functions are constructed via Dubédat's regularized Dirichlet energy. We show that these partition functions are solutions to annulus Belavin-Polyakov-Zamolodchikov (BPZ) equations. In the annulus setup, the number of variables exceeds the number of BPZ equations, so the BPZ system alone does not determine the partition functions uniquely. By establishing sufficiently good control of the two partition functions constructed above, we are nevertheless able to derive the crossing probability.

We study fractional parabolic initial-value problems with lower-order drift and potential terms in anisotropic spectral Barron spaces, defined by weighted space–time Fourier $L^1$ norms adapted to parabolic scaling. We prove existence, uniqueness, and maximal regularity with a gain of one derivative in time and $\gamma$ derivatives in space, where $\gamma>0$ is the order of the fractional Laplacian. The evolution is defined only for $t\geq0$, whereas the finite-time norm requires a global extension with sufficient temporal Fourier decay. We construct a finite reflected semigroup extension using a Vandermonde system to match derivatives at $t=0$, obtaining temporal Fourier estimates uniform in the semigroup parameter. Combined with Fourier multiplier estimates for the damped principal operator, it yields maximal regularity. Dimension-independent multiplication estimates support a finite regularity bootstrap, while interpolation and sufficient damping absorb the lower-order terms in the base estimate. The a priori estimate and the method of continuity yield maximal regularity without smallness assumptions on the lower-order coefficients. A frequency-localized counterexample shows that a uniform-in-time spatial Barron bound on the forcing does not imply the corresponding two-derivative solution bound, even for the one-dimensional heat equation. Using this regularity, Fourier sampling yields $n^{-1/2}$ approximation rates for the solution in mixed space–time Sobolev norms using shallow networks with suitable activations. Sampling in a product Hilbert space yields a population-level PINN consistency estimate for shallow cosine networks on a bounded cylinder. There exists a single width-$n$ network for which the sum of the squared mixed-Sobolev solution error, the squared $L^2$-norm of the residual for the whole-space fractional equation, and the squared initial-data error is $O(n^{-1})$.

In a blogpost in 2009, Gowers raised a possible approach to Littlewood's conjecture in Diophantine approximation, leading to a question about the existence of sufficiently many points in the unit cube such that the “hyperbolic distance" between any two of them is large. In this note we answer this question by an explicit construction. This shows that this approach to prove Littlewood's conjecture cannot work, unless some further refinements are added.

In this paper, we develop a cross-layer end-to-end (E2E) resource orchestration framework for green cell-free massive multiple-input multiple-output (CF-mMIMO) integrated sensing and communication (ISAC) systems with distributed multi-target detection. We propose a distributed sensing approach in which receive access points (RX-APs) compute local test statistics, which are aggregated at the cloud using weights based on sensing interference and channel quality. We derive the local maximum a posteriori ratio test (MAPRT) detectors under fully informed and partially informed operation, representing different levels of transmit-signal information at the RX-APs. We further characterize their processing and fronthaul requirements and derive a network power model incorporating radio transmission, AP and cloud processing, and fronthaul infrastructure. We formulate a mixed-integer non-convex problem that jointly optimizes transmit powers, AP modes, user equipment (UE) and sensing-area associations, RX-AP assignments, and active fronthaul and cloud resources subject to communication, sensing, power, processing-capacity, and fronthaul constraints. A two-stage iterative algorithm based on Big-M reformulation, convex- concave programming, penalty-based relaxation, and structured discrete recovery is developed. Numerical results show that the proposed E2E framework reduces total power by up to 50% compared with transmit-power-only optimization and by approximately 11-17% compared with joint radio optimization under full coordination, while maintaining detection probabilities above 0:95.

Finite logarithms can be defined in the “poor man's adèle ring" $\mathcal{A}$ using Fermat quotients modulo sufficiently large primes. This ring contains $\mathbb{Q}$ and outside the trivial cases, Matsusaka and Seki have shown that finite logarithms cannot take non-zero rational values in $\mathcal{A}$. Furthermore, a theorem of Silverman shows they are not zero, assuming the $abc$-conjecture. We extend these results to primes restricted to arithmetic progressions of the form $p\equiv 1\bmod m$ by relating Fermat quotients to values of cyclotomic polynomials and their logarithmic derivatives. A signed version of the same argument shows unconditionally that the square of a finite logarithm cannot take non-zero rational values in $\mathcal{A}$. As a further application, we show that, subject to the $abc$-conjecture, finite logarithms cannot be quadratic irrational elements of $\mathcal{A}$ in Rosen's theory of finite algebraic numbers.